Solutions to Odd-Numbered Exercises

1. An ellipse is the set of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant.

3. This special case would be a circle.

5. It is symmetric about the x-axis, y-axis, and the origin.

7. yes; [latex]\frac{{x}^{2}}{{3}^{2}}+\frac{{y}^{2}}{{2}^{2}}=1[/latex]

9. yes; [latex]\frac{{x}^{2}}{{\left(\frac{1}{2}\right)}^{2}}+\frac{{y}^{2}}{{\left(\frac{1}{3}\right)}^{2}}=1[/latex]

11. [latex]\frac{{x}^{2}}{{2}^{2}}+\frac{{y}^{2}}{{7}^{2}}=1[/latex]; Endpoints of major axis [latex]\left(0,7\right)[/latex] and [latex]\left(0,-7\right)[/latex]. Endpoints of minor axis [latex]\left(2,0\right)[/latex] and [latex]\left(-2,0\right)[/latex]. Foci at [latex]\left(0,3\sqrt{5}\right),\left(0,-3\sqrt{5}\right)[/latex].

13. [latex]\frac{{x}^{2}}{{\left(1\right)}^{2}}+\frac{{y}^{2}}{{\left(\frac{1}{3}\right)}^{2}}=1[/latex]; Endpoints of major axis [latex]\left(1,0\right)[/latex] and [latex]\left(-1,0\right)[/latex]. Endpoints of minor axis [latex]\left(0,\frac{1}{3}\right),\left(0,-\frac{1}{3}\right)[/latex]. Foci at [latex]\left(\frac{2\sqrt{2}}{3},0\right),\left(-\frac{2\sqrt{2}}{3},0\right)[/latex].

15. [latex]\frac{{\left(x - 2\right)}^{2}}{{7}^{2}}+\frac{{\left(y - 4\right)}^{2}}{{5}^{2}}=1[/latex]; Endpoints of major axis [latex]\left(9,4\right),\left(-5,4\right)[/latex]. Endpoints of minor axis [latex]\left(2,9\right),\left(2,-1\right)[/latex]. Foci at [latex]\left(2+2\sqrt{6},4\right),\left(2 - 2\sqrt{6},4\right)[/latex].

17. [latex]\frac{{\left(x+5\right)}^{2}}{{2}^{2}}+\frac{{\left(y - 7\right)}^{2}}{{3}^{2}}=1[/latex]; Endpoints of major axis [latex]\left(-5,10\right),\left(-5,4\right)[/latex]. Endpoints of minor axis [latex]\left(-3,7\right),\left(-7,7\right)[/latex]. Foci at [latex]\left(-5,7+\sqrt{5}\right),\left(-5,7-\sqrt{5}\right)[/latex].

19. [latex]\frac{{\left(x - 1\right)}^{2}}{{3}^{2}}+\frac{{\left(y - 4\right)}^{2}}{{2}^{2}}=1[/latex]; Endpoints of major axis [latex]\left(4,4\right),\left(-2,4\right)[/latex]. Endpoints of minor axis [latex]\left(1,6\right),\left(1,2\right)[/latex]. Foci at [latex]\left(1+\sqrt{5},4\right),\left(1-\sqrt{5},4\right)[/latex].

21. [latex]\frac{{\left(x - 3\right)}^{2}}{{\left(3\sqrt{2}\right)}^{2}}+\frac{{\left(y - 5\right)}^{2}}{{\left(\sqrt{2}\right)}^{2}}=1[/latex]; Endpoints of major axis [latex]\left(3+3\sqrt{2},5\right),\left(3 - 3\sqrt{2},5\right)[/latex]. Endpoints of minor axis [latex]\left(3,5+\sqrt{2}\right),\left(3,5-\sqrt{2}\right)[/latex]. Foci at [latex]\left(7,5\right),\left(-1,5\right)[/latex].

23. [latex]\frac{{\left(x+5\right)}^{2}}{{\left(5\right)}^{2}}+\frac{{\left(y - 2\right)}^{2}}{{\left(2\right)}^{2}}=1[/latex]; Endpoints of major axis [latex]\left(0,2\right),\left(-10,2\right)[/latex]. Endpoints of minor axis [latex]\left(-5,4\right),\left(-5,0\right)[/latex]. Foci at [latex]\left(-5+\sqrt{21},2\right),\left(-5-\sqrt{21},2\right)[/latex].

25. [latex]\frac{{\left(x+3\right)}^{2}}{{\left(5\right)}^{2}}+\frac{{\left(y+4\right)}^{2}}{{\left(2\right)}^{2}}=1[/latex]; Endpoints of major axis [latex]\left(2,-4\right),\left(-8,-4\right)[/latex]. Endpoints of minor axis [latex]\left(-3,-2\right),\left(-3,-6\right)[/latex]. Foci at [latex]\left(-3+\sqrt{21},-4\right),\left(-3-\sqrt{21},-4\right)[/latex].

27. Foci [latex]\left(-3,-1+\sqrt{11}\right),\left(-3,-1-\sqrt{11}\right)[/latex]

29. Focus [latex]\left(0,0\right)[/latex]

31. Foci [latex]\left(-10,30\right),\left(-10,-30\right)[/latex]

33. Center [latex]\left(0,0\right)[/latex], Vertices [latex]\left(4,0\right),\left(-4,0\right),\left(0,3\right),\left(0,-3\right)[/latex], Foci [latex]\left(\sqrt{7},0\right),\left(-\sqrt{7},0\right)[/latex]

35. Center [latex]\left(0,0\right)[/latex], Vertices [latex]\left(\frac{1}{9},0\right),\left(-\frac{1}{9},0\right),\left(0,\frac{1}{7}\right),\left(0,-\frac{1}{7}\right)[/latex], Foci [latex]\left(0,\frac{4\sqrt{2}}{63}\right),\left(0,-\frac{4\sqrt{2}}{63}\right)[/latex]

37. Center [latex]\left(-3,3\right)[/latex], Vertices [latex]\left(0,3\right),\left(-6,3\right),\left(-3,0\right),\left(-3,6\right)[/latex], Focus [latex]\left(-3,3\right)[/latex]
Note that this ellipse is a circle. The circle has only one focus, which coincides with the center.

39. Center [latex]\left(1,1\right)[/latex], Vertices [latex]\left(5,1\right),\left(-3,1\right),\left(1,3\right),\left(1,-1\right)[/latex], Foci [latex]\left(1,1+4\sqrt{3}\right),\left(1,1 - 4\sqrt{3}\right)[/latex]

41. Center [latex]\left(-4,5\right)[/latex], Vertices [latex]\left(-2,5\right),\left(-6,4\right),\left(-4,6\right),\left(-4,4\right)[/latex], Foci [latex]\left(-4+\sqrt{3},5\right),\left(-4-\sqrt{3},5\right)[/latex]

43. Center [latex]\left(-2,1\right)[/latex], Vertices [latex]\left(0,1\right),\left(-4,1\right),\left(-2,5\right),\left(-2,-3\right)[/latex], Foci [latex]\left(-2,1+2\sqrt{3}\right),\left(-2,1 - 2\sqrt{3}\right)[/latex]

45. Center [latex]\left(-2,-2\right)[/latex], Vertices [latex]\left(0,-2\right),\left(-4,-2\right),\left(-2,0\right),\left(-2,-4\right)[/latex], Focus [latex]\left(-2,-2\right)[/latex]

47. [latex]\frac{{x}^{2}}{25}+\frac{{y}^{2}}{29}=1[/latex]

49. [latex]\frac{{\left(x - 4\right)}^{2}}{25}+\frac{{\left(y - 2\right)}^{2}}{1}=1[/latex]

51. [latex]\frac{{\left(x+3\right)}^{2}}{16}+\frac{{\left(y - 4\right)}^{2}}{4}=1[/latex]

53. [latex]\frac{{x}^{2}}{81}+\frac{{y}^{2}}{9}=1[/latex]

55. [latex]\frac{{\left(x+2\right)}^{2}}{4}+\frac{{\left(y - 2\right)}^{2}}{9}=1[/latex]

57. [latex]\text{Area}=12\pi[/latex] square units

59. [latex]\text{Area}=2\sqrt{5}\pi[/latex] square units

61. [latex]\text{Area }9\pi[/latex] square units

63. [latex]\frac{{x}^{2}}{4{h}^{2}}+\frac{{y}^{2}}{\frac{1}{4}{h}^{2}}=1[/latex]

65. [latex]\frac{{x}^{2}}{400}+\frac{{y}^{2}}{144}=1[/latex]. Distance = 17.32 feet

67. Approximately 51.96 feet